Transverse waves on a string

In the previous three chapters, we built up the foundation for our study of waves. In the remainder of this book, we’ll investigate various types of waves, such as waves on a string, sound waves, electromagnetic waves, water waves, quantum mechanical waves, and so on. In Chapters 4 through 6, we’ll discuss the properties of the two basic categories of waves, namely dispersive waves, and non-dispersive waves. The rest of the book is then largely a series of applications of these results. Chapters 4 through 6 therefore form the heart of this book. A non-dispersive system has the property that all waves travel with the same speed, independent of the wavelength and frequency. These waves are the subject of this and the following chapter (broken up into longitudinal and transverse waves, respectively). A dispersive system has the property that the speed of a wave does depend on the wavelength and frequency. These waves are the subject of Chapter 6. They’re a bit harder to wrap your brain around, the main reason being the appearance of the so-called group velocity. As we’ll see in Chapter 6, the difference between non-dispersive and dispersive waves boils down to the fact that for non-dispersive waves, the frequency ω and wavelength k are related by a simple proportionality constant, whereas this is not the case for dispersive waves. The outline of this chapter is as follows. In section 4.1 we derive the wave equation for transverse waves on a string. This equation will take exactly the same form as the wave equation we derived for the spring/mass system in Section 2.4, with the only difference being the change of a few letters. In Section 4.2 we discuss the reflection and transmission of a wave from a boundary. We will see that various things can happen, depending on exactly what the boundary looks like. In Section 4.3 we introduce the important concept of impedance and show how our previous results can be written in terms of it. In Section 4.4 we talk about the energy and power carried by a wave. In Section 4.5 we calculate the form of standing waves on a string that has boundary conditions that fall into the extremes (a fixed end or a “free” end). In Section 4.6 we introduce damping, and we see how the amplitude of a wave decreases with distance in a scenario where one end of the string is wiggled with a constant amplitude.